Network for Computational Nanotechnology

(NCN)

UC Berkeley, Univ. of Illinois, Norfolk State, Northwestern, Purdue, UTEP

Topological Insulators: Electronic structure, material systems and its applications

Parijat Sengupta

Network for Computational Nanotechnology

Electrical and Computer Engineering

Purdue University

Complete set of slides from my final PhD defense Advisor : Prof. Gerhard Klimeck

A new addition to the metal-insulator classification

2

Metal Trivial Insulator Topological Insulator

Nature, Vol. 464, 11 March, 2010

Surface states

2

Tracing the roots of topological insulator

3

The Hall effect of 1869 is a start point

How does the Hall resistance behave for a 2D set-up under low temperature and high magnetic field (The Quantum Hall Effect) ?

The Quantum Hall Plateau

4

Hall Conductance is quantized

xy ne2

h

h

e2 26k

,...2,1,0 , ,21 n

mc

eBnE ccn

B

Cyclotron motion

Landau Levels

Plateau and the edge states

5

Skipping orbits induce a gapless edge mode along the sample boundary

Landau Levels Skipping orbits along the edge

Ext. B field

Question: Are edge states possible without external B field ?

Is there a crystal attribute that can substitute an external B field

6

Spin orbit Interaction :

1. Momentum dependent force, analogous to B field

2. Opposite force for opposite spins

3. Energy depends on electronic spin

4. Spin-orbit strongly enhanced for atoms of large atomic number

B

Fundamental difference lies under a Time Reversal Symmetric Operation

Internal B field

Time Reversal Symmetry Violation

1. Quantum Hall

2. Intrinsic B field

Quantum Hall effect with spin-dependent B field

7

Two copies of the Quantum Hall Effect

The Quantum Hall effect had skipping orbits induced edge states

What is the counterpart in the case of a spin dependent B field ?

The edge/surface states…

8

Normal Insulator

We need to produce this

Every material possesses these surface states but they are fragile to perturbation and disorder

Robust if they are bound to the surface

Key question : How to generate a bound surface state ?

Step 1) for bound surface states – A zero gap material

9

Zero gap semiconductor

Positive gap semiconductor

*These notations are valid only for ZB crystals

*

Step 2) Inversion of the zero gap material

10

If HgTe is confined – the band gap splits from its zero value to a negative band gap semiconductor

Normal ordered material Inverted ordered material

Bands of opposite parity (p like band and s like band) interchange their relative positions.

Step 3) Contact with a “normal” band-gap ordered material

11

Normal ordered material Inverted ordered material

There must be a state that closes the band-gap (band-gap = 0) for a continuous transition from region I to region II

Interface

Band-gap is > 0 Band-gap is < 0

Region I Region II

E

k

E

k

Experimental observations: Surface states of Bi2Se3

12

Surface state are usually probed using ARPES*

Bi2Se3 has inverted band order at Γ and is in contact with normal-ordered vacuum

k (A-1)

* Angle Resolved Photoemission spectroscopy

The TRIM points

13

Before we proceed, some basic facts about band structure theory 𝐸 𝑘, ↑ = 𝐸 −𝑘 , ↑ Space Inversion symmetry (1)

𝐸 𝑘, ↑ = 𝐸 −𝑘 , ↓ Time reversal symmetry (2)

If both space inversion and time reversal symmetry are simultaneously fulfilled, then the following holds

𝐸 𝑘, ↑ = 𝐸 𝑘 , ↓ Spin degeneracy occurs at the same 𝑘 point (3)

There are certain special points on the Brillouin zone where equation (3) is always true

Such points are known as Time Reversal Invariant Momenta Points (TRIM)

TRIM points

14

On the surface Brillouin zone which is 2D, there are four TRIM points

0,0 , 𝜋, 0 , 0, 𝜋 , 𝜋, 𝜋

Let us pick 𝜋, 0 to prove that it is TRIM Step 1 : Does space inversion symmetry hold ?

𝜋, 0yields

−𝜋, 0 : Separated in 𝑘-space by 2𝜋

Bloch’s theorem tells that points separated by 2𝜋 are equivalent

∴ 𝜋, 0 and −𝜋, 0 are space inversion symmetric

Time reversal symmetry holds in general 𝜋, 0 is TRIM!

How does the band structure appear at the TRIM points ?

15

Energy bands are time reversed pairs which are degenerate at 𝑘 = 0 and 𝑘 = 𝜋

This degeneracy is a fundamental degeneracy that cannot be removed!

Trivial Insulator Topological Insulator

Energy bands are time reversed pairs which are degenerate at 𝑘 = 0 and 𝑘 = 𝜋

Time reversed pairs of conduction and valence energy bands also meet at 𝑘 = 0 since it is a point of inversion : Fundamental degeneracy !

Topology

16

The key concept is of smooth deformation :

Sphere can be transformed in to an cuboid

A coffee mug in to a donut

But a sphere cannot be transformed in to a donut without scooping material out of it!

Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything.

Sphere and donut belong to two different classes

Topology and electron bands

17

Cond band

Valence band

Band-gap > 0 is preserved

Smooth deformation

Band-gap changes from + to -

Not a smooth deformation

Positive band-gap and negative band-gap material belong to two different topological classes!

Topology and electron bands

Cond band

Valence band

A B A B Band-gap closing

Two topologically inequivalent classes A and B cannot be continuously deformed in to one another. A connection state must be present.

Must proceed as

Which compounds exhibit topological insulator behaviour?

18

Naturally available topological insulators

19

Bi2Te3 , Bi2Se3 , Sb2Te3 are topological insulators: Robust states on the surface

under room temperature conditions

qu

intu

ple

laye

r

Te Bi Te Bi Te

Bi2Te3

Inequivalent Te atom serves as centre of inversion

Quintuple layer ordered as Te(1) - Bi(1) -Te(2) - Bi(1) - Te(1)

These belong to the class of 3D topological insulators

Engineered topological insulators

20

Topological insulators with protected states on the edge of a quantum well

CdTe/HgTe/CdTe quantum well heterostructure (zinc-blende)

GaSb/InAs quantum well heterostructure (zinc-blende)

These are known as 2D topological insulators

Numerical modeling of 3D topological insulators

Dispersion obtained through ARPES*

*Rev of Mod. Phys, Vol 82, Oct-Dec 2010

sp3d5s* tight binding calculation

Six quintuple layer (~6.0 nm) Bi2Te3 quantum well

Stable surface states (SS) in the mid-gap region

Surf. states

21

Validation: Reproducing experimental Fermi surface warping

22

Ener

gy [

eV]

k [nm-1]

[eV]

Fermi surface

Similar warping of iso-energetic Bi2Te3 surfaces has been observed in ARPES measurements.

Numerical calculations performed with 20-band tight binding model

Surface states in a six quintuple layered quantum well (6.0 nm)

23

Bi2Te3 Bi2Se3 Sb2Te3

Band structure of 3D topological insulators with 4-band k.p model*

How do k.p calculations compare to tight binding?

Hamiltonian for the surface state

24

The surface states are characterized by a linear Hamiltonian given as The next higher order term that satisfies C3v symmetry must be k3 What this means is that as we move away from the Γ point, higher order terms dominate Experimentally proven through ARPES*

( )f x y y xv p p

3 3( ) ( )f x y y x zv p p k k

*Physical Review Letters, 104(1):016401, Jan 2010

Comparing k.p to tight binding for 3D TIs

25

[eV]

Fermi surface

sp3d5s* tight binding gives correct Fermi-surface warping Full symmetry (including higher order terms) included in the Hamiltonian

k.p can correctly predict energies close to the Γ point. Full crystal symmetry absent!

Energy contour with modified k.p that includes higher order k3 terms

Energy contour with 20-band tight binding

Common electronic structure trait in 2D and 3D TIs

26

Both 2D and 3D TIs are narrow band-gap compounds with high spin-orbit coupling

Spin-orbit coupling induces band inversion necessary for a TI

Question : Any other intrinsic mechanism like so-coupling that can invert bands?

Broken gap devices yield inverted band order

Is there any other crystal attribute that can invert bands?

27

Spin-orbit coupling which inverts bands is effectively an internal magnetic field

Question: Can electric field similarly invert bands?

Commonly used wurtzite-based nitride crystals are moderate ferroelectrics Possesses an internal polarization (electric) field : A crystal property

Wurtzite belongs to hexagonal crystal lattice system with two inter-penetrating hcp lattices

c/a = (8/3)1/2 This ratio has curious properties

Spontaneous polarization in Wurtzite crystal

Unit Cell of Wurtzite XN

Purple = X (Al, Ga, In) Green = N

• In ideal wurtzite crystal the c/a ratio = (8/3)1/2

• In real wurtzite crystal this ratio is altered

- The X-N (X = Al, Ga, In) bond along c-axis is deformed leading to a net dipole moment

• The dipoles cancel in bulk but remain at the surface

Spontaneous Polarization due to crystal asymmetry in wurtzite

28

Schematic showing operation of internal E-field in nitrides*

*Journal of Applied Phys. Vol.85, 3222 (1999)

1. To invert the band order internal electric field must be maximized

2. Piezoelectric field is stronger than spontaneous polarization

3. A strong piezo field needs a large strain

4. InN grown on GaN has large strain (~11 % lattice mismatch)

5. InN is a narrow band-gap material

29

The GaN-InN-GaN system

GaN InN

0.5 eV

2.2 eV

0.7 eV

Ev = 0

Ec = 3.4 eV

Straddling : Type-I heterojunction

GaN and InN are lattice mismatched by ~11 %

Large piezoelectric constants result in strong internal electric field

Spontaneous polarization is closely matched Less significant

Ga-faced

GaN

InN

Psp

Psp Ppz

GaN Psp

GaN is relaxed. InN is biaxially strained (compressive)

Spontaneous and piezoelectric polarization field

Potential drop in InN well = 3.827 V

Fixed charge potential

GaN/InN/GaN produces large built-in fields

30

What can this large electric field do?

The large electric field alters the position of the band edges

For instance: Gradient of band edges due to built-in electric field of a p-n junction

Vbi

Ec

Ef

Ev

p-type

n-type

Similar trend for the band-edge position of the InN region

Ec1

Ev1

Ec1

Ev1

Coordinate position of InN well

The conduction band edge has fallen below the valence band edge

Creation of a broken-gap heterostructure

Potential drop in InN well

31

8-band k.p electronic structure calculations

Electric field outside the structure is set to zero

4

GaN

InN

GaN 1

2

3

1E

2E

3E

1 1 1E

2 2 1 1 2E E

1 3 2 2 3E E

1 sp GaN

2 sp sp pzGaN InN

3 2

4 1

Strain is computed with the Bir-Pikus def. potentials

and are dielectric const. of GaN and InN respectively 1 2

Geometry Construction

Structure Relaxation

Strain

Single Particle Energies

Eigen Solver

Electrostatics

Hamiltonian Construction

8 8x sp pz strainH H V H

H = Eff. Hamiltonian

= Interface charge i

32

Band profile of the GaN/InN/GaN heterostructure

In a normal-ordered nitride in wurtzite phase : Γ1 is energetically placed over Γ6 at the Γ point

Inverted band-order

Mid-gap linearly dispersing states Signature of a Topological Insulator

Topological insulator states obtained by constructing a nano-ribbon of width 100 nm from inverted quantum well on left

GaN/InN/GaN well GaN/InN/GaN nano-ribbon

33

Wave function of the surface states

Mid-gap states localized at the surface Spin-polarization of bands along z-axis

Zero spin-component along x and y axes. Spin perpendicularly locked to momentum

34

Another geometry dependent TI is CdTe/HgTe quantum well

Normal Inverted

- Quantum well heterostructure (QW) is truncated to form a ribbon

- Bound states exist on the ribbon/edge

35

The CdTe/HgTe/CdTe quantum well

36

CdTe/HgTe/CdTe quantum well heterostructure is a 2D topological insulator:

HgTe has inverted band structure and placed adjacent to CdTe which is normal

band order

Bulk band structure of HgTe and CdTe

E1 < H1

E1 : Electron Band ; H1 : Hole Band ; w : Quantum well width

Phase Transition in 2D TI (8 band k.p calculations)

Zero-Gap

E1 = H1

w = 6.3 nm w > 6.3 nm

E1 < H1

w < 6.3 nm

Normal

E1

H1

E1 > H1

E1 H1

H1 E1

Inverted

37

Inverted bulk gives edge/surface states

Gapped states Gapless states

Amplitude of edge wave

function decays in the bulk

38

Typical dimensions of a CdTe-HgTe 2D TI

HgTe well Highest val. band Lowest cond. band

dQW < dC H1 E1

8.2 nm > dQW > dC E1 H1

dQW > 8.2nm H2 H1

• Well composition

• Changeable barrier potential

• Temperature

• Electric field

• Effective masses of electrons and holes

Is the critical width tunable under external influences?

39

Critical Width as a function of well region

CdxHg1-xTe

CdTe CdTe

CdxHg1-xTe serves as well material

Critical width dependent on temperature and mole fraction

Temperature dependent band gap

modulates the critical width

Increasing critical width with mole fraction

40

How to reduce the critical width with barrier material

HgTe

CdxHg1-xTe CdxHg1-xTe

Barrier Critical

Width (nm) Reduction

CdHgTe 6.3 Standard

Cd0.92Hg0.08Te 5.8 7.94 %

Cd0.84Hg0.16Te 5.4 14.29 %

Cd0.76Hg0.24Te 5.0 20.63 %

Cd0.68Hg0.32Te 4.4 28.57 %

Critical width is reducible up to 4.4 nm

with proper barrier material

41

Reduction of critical width with effective mass

Coupling can be tuned through variable

eff. hole and electron masses

Eff. mass can be changed with crystal growth direction

Growth Axis Critical Width Reduction

<100> 6.3 nm Standard

<110> 5.72 nm 9.2 %

<111> 5.52 nm 14.13 %

<111> growth axis provides least critical width

42

Critical width with appl. of strain under diff. growth axis

Tensile Uniaxial Stress

Growth Axis <001> <110> <111>

<001> L H I

<110> L H I

<111> H L I

Comp. Uniaxial Stress

Growth Axis <001> <110> <111>

<001> L H I

<110> L H I

<111> H L I

L = Least; H = Highest; I = Intermediate

43

Impact of external electric field

E field Electric field induced TI

Critical width < 6.3 nm

with increasing E field

dw

CdTe/HgTe heterostructure acts as a

topological switch under electric field

Can such TI-based electronic components

incorporated in a fast switching environment? 44

A fast switching environment needs faster transmission

Material Mobility(cm2 /V.s) Method Reference

Bi2Te3 (bulk) 10,200 Hall meas. Science 329, 821 (2010)

Bi2Se3 (bulk) 13,200 SdH Nano Lett.,12 (3), 1486 (2012)

Bi2Te3 (NR) 4,560-5,790 SdH Nature Nanotech 6,216 (2011)

Topological insulators are characterized by high electron mobility on the surface

Silicon electron mobility is 1400 cm2 /V.s at room temperature (300 K)

not

allowed

Back Scattering is forbidden

45

TIs have high mobility: Is a real fast transistor possible ?

46

Two TI attributes are useful:

1. They have metal-like surface states

2. Surface states possess high mobility

z

x

• Geometric confinement:

8.942 nm (z-axis)

• Open boundary condition

(transport direction): 1.972

nm (x-axis)

• Periodic direction : y axis

• Temperature of operation :

4K / 300K

• Chem. potential of drain = 0

eV relative to band structure

• Chem. potential of source =

[voltage] eV relative to band

structure

• Flat bands

Simulation structure for Bi2Te3

ultra-thin-body

Id vs. Vd characteristics of Bi2Te3

EfS = 0.15 eV

EfD = EfS - Vds

Nominal externally applied voltage between source and drain

results in a sufficiently high current

47

Turning off the TI transistor

The TI transistor has metal-like conduction at surface with high mobility!

But gapless surface states mean No electrical tuning of Fermi level in a band

gap region No “turn-off” possible

A band gap can be induced in a TI in three possible ways

1. Doping with a ferromagnet (FM)

2. Coating with a ferromagnet

3. Superconductor in proximity of a TI

ARPES* measurements

of Cr based Bi2Se3

Gapped surface states of

a TI grown on a FM* *PSS, RRL 7 (2013) 48

The TI-superconductor heterostructure

TI grown

epitaxially

on a SC

SC film on

top of a TI

film

Band gap created on TI surface

Explanation lies in the proximity effect between a superconductor and the TI’s surface state

Cooper pair

TI SC

49

Cooper pairs: The microscopic origin of superconductivity

50

• There is no one single theory that describes superconductivity

• Common to all superconductors is the pairing of electrons that allows dissipation-less transport

• Several mechanisms (not yet fully understood) can give rise to electron-pairing

• The well-accepted (experimentally verified) BCS* theory explained superconductivity at temperatures close to absolute zero for elements and simple alloys

*Bardeen, Cooper, Schrieffer Nobel Prize (Physics) 1972

Paired electrons are called Cooper pairs

Cooper pairs within a lattice

The BCS ground state

The most distinctive consequence of BCS theory is that an energy gap opens up between the ground state and the lowest excited state

The ground state of the superconductor is a linear superposition of states in which pairs (k↑,−k↓)are occupied or unoccupied.

The ground state of the wave function is a superposition of states in which pair are occupied or unoccupied

( , )k k

51

Proximity effect of a superconductor

52

• The Cooper pair ordering of electrons in superconductor is observed in a non-superconductor placed in contact

• Electrons cannot lose their ordering abruptly Coherence is lost over a distance

• Gapped state of the superconductor is induced in the adjacent contact

• Most extensively studied are metal-superconductor junctions

The same proximity effect is visible in a topological insulator-superconductor junction

TI SC SC* TI

Hatched region is TI + SC

*s-wave superconductor

Participating orbitals in surface state bands

Stage 1:Bi and Se

atoms combine to

produce the P1 and P2

orbitals

Stage 2: Crystal field

splitting : Low-energy

sub-space

Stage 3: Spin-orbit

coupling

The four low-lying states at Γ are and

53

Writing the composite TI-superconductor Hamiltonian

54

The Hamiltonian for a superconductor cannot be solved analytically An approximation known as the Bogoliubiv de-Gennes is used to solve the Hamiltonian The general form of the Hamiltonian is is the Hamiltonian of the system without superconductivity (4-band model for TI) is the order parameter, the Fermi-level and is the time reversal operator

* 1

hH

ThT

h

T

Effectively turned in to 8N x 8N sized Hamiltonian

1zP , are paired by Δ1 and , are paired by Δ2 1zP 2 zP 2zP

Superconductivity induced band gap on surface of a 3D TI

55

0.1349 eV

0.1344 eV

A gap of 0.5 meV induced on surface of TI

A s-wave superconductor and 8 quintuple layer (8.0 nm) thick Bi2Te3 heterostructure

Band gap induced is of the same order as the superconducting gap

Conclusion

1. General idea of metals, semiconductors, and insulators

2. New material topological insulator (TI) : Dual properties of metal and insulator

3. Robust protected surface states : 2-D and 3-D topological insulators

4. Electronic structure calculation for TIs: 20 band TB and k.p

1. Internal polarization induced 2D-TIs – Unconventional since it does not use so-

coupling

2. Design principles for HgTe based 2D-TIs – Tunability of the critical width

3. Topological insulator based switching

1. I-V characteristics of Bi2Te3 UTB – No band gap prevents turn-off

2. Inducing a band gap on surface of a 3D-TI TI-Superconductor heterostructure

56

Electronic structure calculation for broken-gap devices P. Sengupta, T. Kubis, Y. Tan, M. Povolotskyi, and G. Klimeck

Objective

• Accurately model the eigen spectrum of

broken-gap devices and topological

insulators Applicable when classical

electron hole description fails

Method

• Consider all states as electronic and

assume a positive charge background

• Screening of core shell electrons is

neglected (ε = 1)

• Total charge density per atom = - electron

density per atom + ionic charge

• To compute positive background charge:

Calculate bulk density in TB

• Electron density at each atom = positive

ionic charge

• Bulk parameters are transferred to

heterostructures

• Self-consistently solve the Schrodinger-

Poisson equation with bulk parameters

Result

• Charge self-consistent tight-binding

calculation done for a Bi2Te3 thin-film

• Energy separation between Dirac cones

gets enhanced

• Fermi velocity of Dirac states changes 57

3D-TI based nanowires P. Sengupta, T. Kubis, Y. Tan, M. Povolotskyi, and G. Klimeck

Objective

• Gain understanding of surface states (SS)

on cylindrical topological insulator (TI)

nanowires

• Influence of B-field on SS dispersion

• Study A-B oscillations on TI nanowires

Method

• 20-band tight binding and 4-band k.p

Hamiltonian

• B-field implemented with the Peierls phase

and Peierls substitution for TB and k.p

respectively

Results

• Diameter of TI nanowires must be more

than 30 nm to observe SS

• Wires grown along z-axis do not have TI

states

• Wires grown along z-axis show band-gap

closing under half magnetic flux quantum

Band-gap closing under half-

magnetic flux quantum Small diameter nanowires lead

to surface volume hybridization

No TI states found

58

59

Numerical study of surface states of 3D-TI materials P. Sengupta, T. Kubis, Y. Tan, M. Povolotskyi, and G. Klimeck

Objective

• Determine optimal conditions for existence

of surface states (SS) on a 3D-TI

• Control the spin polarization of SS by

placing two 3D-Tis together

Method

• 4-band k.p Hamiltonian for Bi2Se3 and

Bi2Te3

• Bias added to Hamiltonian to mimic

asymmetric surface terminations

• Spin-polarizer device parameters analyzed

using 2D-Dirac equation

Results

• Physical and chemical attributes of TI thin-

film impact the SS dispersion

• Low film thickness and asymmetric growth

conditions form displaced Dirac-hyperbolas

• Spin-polarized components have unequal

strength in asymmetric film

59

Sponsors and Funding agencies

• Intel Corporation:

• National Science Foundation: Grant Nos.

EEC-0228390, OCI-0749140

• Semiconductor Research Corporation: Task 2141

• Computational resources at nanoHUB.org: